Option Trading Blog

I stumbled upon Peter Carr’s site and saw a very nice explanation on the arbitrage principles that govern option pricing. Usually explanations on option pricing are very cumbersome and not clear. Here is my take on it, but you can just go to his website and find it in the research papers under FAQ.

Suppose you have a stock that is priced at 1$ and it has a 50% on either going up to 2$ or going down to 0.5$. Thus, we assume there are only two states with equal probablity. Let us now assume there is a Call option written on that stock with strike price of 1$. Thus, in the up state it makes 1$ and in the down state it makes 0. Now comes the million dollar question.. how much is the option worth? If you take the expected value approach, you’d say it should be worth 0.5$ since that is the expected profit ()

There are a few problems with the expected value approach. We could also say that the stock should be worth 1.25$ in the future by the same expected value approach. This adds more confusion to the proper way to price the option.

Let us assume that we go with the first approach and we price it to be worth 0.5$. In that situation we could sell two call options and buy one stock. We have no initial cost. If the stock goes up, we make nothing, but if the stock goes down, we profit 0.5$ since the options expire worthless. That is in essence an arbitrage, since we have no risk at all and we stand to gain a profit (although in only one state of the world). This tells us that something is wrong with this approach. The question is, is their a proper way of doing this?

Let us assume that there is no interest in this world of ours for simplicity. This means that as time passes, we demand no compensation for our money if we take no risk. We know that the option pays 1$ in the upstate, and nothing in the downstate. We can construct a portfolio that replicates the option buy noticing that buying 2/3 of the stock and selling short 1/3 of a bond with face value of 1$ with exactly replicate the option price (do it yourself to make sure). The total cost would be 1/3$. What is that magical 1/3$ number we got? I claim that is should be the fair price of an option!

As a good exercise, you should try to figure out why this should be the fair price. But do not worry. In the next post, I’ll explain more about why this should be the fair price and touch upon what is known as state prices.

One of the most important tools for the option trader are the Parkinson number and the Variance Ratio Method. The Parkinson number, named after physicist Michael Parkinson, is an estimator of the volatility of returns presupposing that returns follow a geometric random walk (If that last bit doesn’t tell you anything, don’t worry).

If you take a look at the last post when we estimated volatility, we needed to take a sample of each day/week/month we wanted to estimate. The Parkinson method estimates the volatility only by the high and low of any particular period. Following is the formula. Don’t worry if it looks scary, I’ll walk you through it:

Where and are the High and Low for that period respectively.

The meaning

If this number is supposed to give us the estimated volatility, we can compare it to the sampling formula we derived in the post about volatility estimation. This gives us the following relation:

Where is the sampled volatility we estimate.

This last equation tells us that the Parkinson number will be 1.67 times bigger than the sampled volatility.

Why Should You Care

Who cares really you ask? We are interested in the cases when the Parkinson number is either bigger or lower than 1.67 times the volatility. When The Parkinson Number is higher than 1.67 times the volatility, it means that the markets are volatile and one should follow a trend. This is just one application. You can think of many more applications when you know volatility is high.

Another application is with hedging. If the Parkinson Number is lower, than we can assume the market is less volatile and hence, perhaps not re-balance our hedge to keep ourselves market neutral.

I’ll soon add an excel file so you could implement this on your own. I’ll also add some application with the Parkinson number so you could see how good of an indicator it is. Also, in the next post, I’ll discuss the Variance-Ratio Method and how it can help you profit in your trades.